Channel codes, sometimes called modulation codes, are mappings of data bits into the symbols that are either transmitted in a communication system or recorded onto a medium in a storage device. The purpose of these codes is to prevent certain characteristics in the stream of symbols that make their recovery difficult. Runlength limited (RLL) codes are commonly used in magnetic recording. These codes impose a (d,k) constraint on the recorded data sequence. With the Non-Return-to-Zero (NRZ) recording format, where the binary "1" represents a positive level in the magnetization waveform and the binary "0" negative level in the same waveform, d+1 is the minimum number of consecutive like symbols and k+1 is the maximum number of consecutive like symbols in the binary sequence. With the Non-Return-to-Zero-Inversion (NRZI) recording format, where a magnetic transition is represented by 1 and no transition by 0, d and k are the minimum and maximum number of consecutive 0's between any two 1's, respectively as described in P. H. Siegel, "Recording codes for digital magnetic storage," IEEE Transactions on Magnetics, vol. MAG-21, no. 5, pp. 1344-1349, September 1985. The d constraint is used to increase the minimum physical spacing between transitions. The k constraint guarantees that a change in the readback waveform will occur at regular intervals for the purpose of synchronizing a phase locked loop to the data. A (1,7) code is a common example of an RLL code; see U.S. Pat. No. 4,337,458. Also popular is the (0,4/4) code, where d=0 and k=4 both for the data sequence and for the sequence that results if every other symbol is considered; see U.S. Pat. No. 4,707,681. Additional constraints, such as a limitation on the total number of NRZI 1's in a codeword for the purpose of improving timing and gain control can be applied to these codes; see U.S. Pat. No. 5,196,849. A DC-free constraint as described in U.S. Pat. No. 4,499,454 can be used to reduce the low frequency spectral content of the readback signal. Codes for data storage typically assume a binary symbol set such as the polarity of the write signal or the presence and absence of a transition, but it is possible to conceive systems that use more than two distinct symbols. For example, the ternary 3PM code uses three distinct symbols and places a lower bound on the distance between symbols in the same way that the RLL d constraint is applied to the binary case. See G. V. Jacoby, "Ternary 3PM magnetic recording code and system," IEEE Transactions on Magnetics, vol. MAG-17, no. 6, pp. 3326-3328, November 1981. In optical data storage, a special type of RLL constraint is applied to guarantee the minimum size of the written mark on the medium as described in R. Karabed and P. H. Siegel, "Even mark modulation for optical recording," International Conference on Communications, June 1989. While RLL (1,k) coding has many useful properties, the required code rate, given by the number of data bits per channel bit, is typically low, forcing the channel to operate at a considerably higher speed than the actual data rate. On the other hand, (0,4/4) or more generally (0,G/I) coding offers a much higher rate, but does not provide any coding gain. Also, (0,G/I) codes are designed specifically for interleaved systems such as class IV partial response (PR4) systems, and are not optimal for other detectors such as fixed-delay tree search (FDTS) systems.
Sequence detectors are data recovery devices that examine multiple received samples to recover the input data sequence. Methods such as Viterbi detection, FDTS/DF, and PRML are all sequence detectors. In magnetic data storage devices, the response of the channel to an input symbol typically extends over several sample periods. Sequence detectors can outperform sample-by-sample decision rules such as peak detection by using information about the data to be detected contained in adjacent samples. Errors in sequence detectors arise mostly from difficulty in distinguishing minimum distance patterns. For a sequence detector that uses M samples to make a decision, all possible noiseless sample sequences can be plotted as points in an M-dimensional space, where each sample corresponds to a coordinate in this space. The minimum distance patterns are those patterns corresponding to different decisions that have the minimum Euclidean distance from one another. The Euclidean distance is the geometric distance between two points and refers to the square root of the sum of the squares of the differences between the coordinates of two points. The performance of sequence detectors such as E.sup.2 PRML can be improved by coding to remove the patterns that cause minimum distance error events, thereby increasing the minimum distance. This increase in the minimum distance as a result of coding is termed coding gain. See R. Karabed and P. H. Siegel, "Coding for higher order partial response channels," Proceedings of the International Society for Optical Engineering, vol. 2605, pp. 115-126, 1995.